

which are derived from the commutation relation \([\hat{x},\hat{p}]=i\)  and the Hadamard´s lemma [1]}, [2]}, [3]} \(e^{g\hat{A}}\hat{B}e^{-g\hat{A}}=\hat{B}+g[\hat{A},\hat{B}]+\frac{g^2}{2!}[\hat{A},[\hat{A},\hat{B}]]+\dots \)  when \(g=\frac{r}{2}\) ; we obtain
\(\hat{\mathcal {H}}=\exp \left(2ir\right) \hat{p}^2+\exp \left[ -ir\left( 2+\epsilon \right)+i\frac{\pi }{2}\epsilon \right] \hat{x}^{2+\epsilon }.\) 
